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By Nicolas Privault
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Extra info for Notes on Stochastic Finance
Sample text
Also, ξ¯0 does not exist and is not needed in this framework. Of course the chosen unit of time may not be the day, and it can be replaced by weeks, hours, minutes, seconds, etc. We will denote by Vt := ξ¯t · S¯t the value of the portfolio at time t = 1, . . , N , with V0 = ξ¯1 · S¯0 . Let ¯ t = (Xt(0) , Xt(1) , . . , Xt(d) ) X denote vector of discounted asset prices defined as: (i) Xt = 1 (i) S , (1 + r)t t or ¯ t := X i = 0, 1, . . , d, 1 S¯t , (1 + r)t t = 0, 1, . . , N, t = 0, 1, . . , N.
Let the centered and normalized return Yt be defined by a−r , b−a Rt − r = t = 1, . . , N. Yt = b−a b−r , b−a Note that we have IE[Yt ] = 0 and var [Yt ] = pq, t = 1, . . , N, and that the discounted asset price increment reads 1 Xt−1 (Rt − r) 1+r b−a Yt Xt−1 , t = 1, . . , N. = 1+r Xt − Xt−1 = The next proposition is the Clark predictable representation formula, cf. g. 1. 3. html Notes on Stochastic Finance ∞ IE[Dt F |Ft−1 ]Yt . 11) t=1 In addition to the Clark formula we also state a discrete-time analog of Itˆo’s change of variable formula, which will be useful for option hedging, cf.
N. = 1+r Xt − Xt−1 = The next proposition is the Clark predictable representation formula, cf. g. 1. 3. html Notes on Stochastic Finance ∞ IE[Dt F |Ft−1 ]Yt . 11) t=1 In addition to the Clark formula we also state a discrete-time analog of Itˆo’s change of variable formula, which will be useful for option hedging, cf. 1 of [39]. 4. Let (Mn )n∈N be a martingale and f : R × N → R. 12) k=1 t (IE[f (Mk , k)|Fk−1 ] − f (Mk−1 , k − 1)) . e. 13) t = 1, . . , N . On the other hand, the term IE[f (Mt , t) − f (Mt−1 , t − 1)|Ft−1 ] is analog to the finite variation part in the continuous time Itˆo formula, and can be written as pf Mt−1 + ut q , t + qf p Mt−1 − ut p , t − f (Mt−1 , t − 1) .